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\begin{document}

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\newcommand{\tit}{\bf Design and Analysis of the Randomized Response
  Technique}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\if0\blind

{\title{\tit\thanks{The proposed methods presented in this paper are
      implemented as part of the R package, {\tt rr: Statistical
        Methods for the Randomized Response Technique}
      \citep{blai:zhou:imai:15}, which is freely available for
      download at
      \href{http://cran.r-project.org/package=rr}{http://cran.r-project.org/package=rr}.
      The replication archive is available as~\citep{blai:imai:zhou:15b}.
      The survey research was approved by the Princeton University
      Institutional Review Board under Protocol \# 5350 and was
      supported by the International Growth Centre (RA--2010--12--013;
      Blair and Imai).  Zhou acknowledges support from the National
      Science Foundation (SES--1148900).}  }

  \author{Graeme Blair\thanks{Pre-Doctoral Fellow, Experiments in Governance and Politics, Columbia University. Email:
      \href{mailto:graeme.blair@columbia.edu}{graeme.blair@columbia.edu}, URL:
      \url{http://graemeblair.com}} 
    \hspace{.5in} Kosuke Imai\thanks{Professor, Department of Politics, Princeton University,
      Princeton NJ 08544. Phone: 609--258--6601, Email:
      \href{mailto:kimai@princeton.edu}{kimai@princeton.edu}, URL:
     \url{http://imai.princeton.edu}} 
   \hspace{.5in} Yang-Yang Zhou\thanks{Ph.D. student, Department of Politics, Princeton University, Princeton NJ 08544. Email:
      \href{mailto:yz3@princeton.edu}{yz3@princeton.edu}, URL:
      \url{http://yangyangzhou.com}}}

  \date{First Draft: September 29, 2014\\
This Draft: \today
}

<<label = setup, cache = FALSE, include = FALSE>>=
opts_chunk$set(eval = TRUE, echo = FALSE, message = FALSE, error = FALSE, warning = FALSE, results = 'hide', cache = FALSE)
@

<<label=packages, cache = FALSE>>=

library(rr)
library(xtable)

data(nigeria)

logistic <- function(object) exp(object)/(1+exp(object))
@

<<label = powerComparisonFunction, cache = FALSE>>=
set.seed(1)

p.vals <- seq(from = .0001, to = .9999, by = .001)
p1.vals <- seq(from = .0001, to = .9999, by = .001)

power.plot <- function(presp = .1, n = 1000, pv = p.vals, p1v = p1.vals, 
                       p1 = NA, p0 = NA, p = NA, design = "forced-known", vary = "p") {          
  
  has.p1 <- !is.na(p1)
  has.p0 <- !is.na(p0)
  
  if(vary == "p") {
    vals <- pv
    if(design == "forced-known" | design == "unrelated-known")
      vals <- vals[vals < (1-sum(p1, p0, na.rm = T))]
    else if(design == "mirrored")
      vals <- vals[vals!=.5]
  } else if(vary == "p1") {
    vals <- p1v
    vals <- vals[vals < (1-p)]
  }
  
  return.list <- rep(NA, length(vals))    
  for(v in vals) {
    val.num <- which(round(vals, 4) == round(v, 4)) 
    print(val.num)
    
    if(vary == "p") {
      if(!has.p1) { 
        p1 <- 1-v-p0
      } else if(!has.p0) {
        p0 <- 1-p1-v
      }
      
      return.list[val.num] <- power.rr.test(p = v, n = n, p1 = p1, p0 = p0,
                                            presp = presp, presp.null = 0,
                                            design = design, sig.level = .05, 
                                            type = "one.sample",
                                            alternative = "one.sided")$power
    } else if (vary == "p1") {
      if(has.p0)
        stop("when varying p1 don't set p0")
      
      p0 <- 1-v-p
      
      return.list[val.num] <- power.rr.test(p = p, n = n, p1 = v, p0 = p0,
                                            presp = presp, presp.null = 0,
                                            design = design, sig.level = .05, 
                                            type = "one.sample",
                                            alternative = "one.sided")$power
    }
  }
  return(cbind(vals, return.list))
}
@

\maketitle
}\fi

\if1\blind \title{\bf \tit} \maketitle
\fi

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\pdfbookmark[1]{Title Page}{Title Page}

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\begin{abstract}
  About a half century ago, Warner (1965) proposed the randomized
  response method as a survey technique to reduce potential bias due
  to non-response and social desirability when asking questions about
  sensitive behaviors and beliefs.  This method asks respondents to
  use a randomization device, such as a coin flip, whose outcome is
  unobserved by the interviewer.  By introducing random noise, the
  method conceals individual responses and protects respondent
  privacy.  While numerous methodological advances have been made, we
  find surprisingly few applications of this promising survey
  technique.  In this paper, we address this gap by (1) reviewing
  standard designs available to applied researchers, (2) developing
  various multivariate regression techniques for substantive analyses,
  (3) proposing power analyses to help improve research designs, (4)
  presenting new robust designs that are based on less stringent
  assumptions than those of the standard designs, and (5) making all
  described methods available through open-source software.  We
  illustrate some of these methods with an original survey about
  militant groups in Nigeria.
  
\bigskip
\noindent {\bf Key Words:} power analysis, randomization, sensitive
questions, social desirability bias

\end{abstract}

\clearpage
\spacingset{1.85}

\section{Introduction}
\label{sec:intro}

About a half century ago, \citet{Warner:1965} proposed the randomized
response method as a survey technique to reduce potential bias due to
non-response and social desirability when asking questions about
sensitive behaviors and beliefs.  The method asks respondents to use a
randomization device, such as a coin flip, whose outcome is unobserved
by the interviewer. Depending on the particular design, the
randomization device determines which question the respondent answers
\citep{Warner:1965, Greenberg:1969, Mangat:1990, Mangat:1994}, the
type of expression the respondent uses to answer the sensitive
question \citep{Kuk:1990}, or if the respondent should give a
predetermined response \citep{Boruch:1971, Fox:1986}. By introducing
random noise, the randomized response method conceals individual
responses and protects respondent privacy. As a result, respondents
may be more inclined to answer truthfully.

Despite the wide applicability of the randomized response technique
and the methodological advances, we find surprisingly few
applications.  Indeed, our extensive search yields only a handful of
published studies that use the randomized response method to answer
substantive questions \citep{Madigan:1976, Chaloupka:1985,
  Wimbush:1997, Donovan:2003, StJohn:2012}.  In contrast, a vast
majority of existing studies apply the randomized response method in
order to empirically illustrate its methodological properties by
including some substantive examples \citep[e.g.,][]{Abernathy:1970,
  Chi:1972, Goodstadt:1975, Reinmuth:1975, Locander:1976, Fidler:1977,
  Lamb:1978, Tezcan:1981, Tracy:1981, Edgell:1982, Volicer:1982,
  Heijden:1996, Heijden:2000, Elffers:2003, Lensvelt:2005b, Lara:2006,
  Cruyff:2007, Himmelfarb:2008, Gingerich:2010, Dejong:2010,
  Krumpal:2012}.  This finding is consistent with previous reviews of
the literature.  Like \citet{Umesh:1991}, a recent review by
\citet{Lensvelt:2005a} concludes that ``there have been very few
substantive applications of RRTs [randomized response techniques] and
that most papers are published to test a variant or illustrate a
statistical problem'' (p. 325).

In this paper, we fill this gap by providing a suite of methodological
tools that facilitate the use of randomized response technique in
applied research.  We begin by reviewing and comparing the standard
designs available to researchers (Section~\ref{sec:Designs}).  We
categorize commonly used designs into four basic groups and discuss
identification and practical issues by using examples from existing
studies.  Building on the results in the literature, we then develop
various multivariate regression techniques for substantive analyses
(Section~\ref{sec:Statistics}).  In particular, we show how to use the
randomized response as a predictor as well as the outcome in
regression models.  We also propose power analyses to help improve
research designs and discuss the pros and cons of each design from a
practical perspective.  Using an original survey about militant groups
in Nigeria, we illustrate some of these methodologies
(Section~\ref{sec:Application}).

The dearth of substantive applications is unfortunate because there
exists empirical evidence that the randomized response method is an
effective technique for studying sensitive topics at least in some
settings \citep[e.g.,][]{Tracy:1981, Heijden:2000, Lara:2004,
  Rosenfeld:2014, Stubbe:2014}.  Some researchers find that not only
does randomized response lead to the lowest response distortion
compared to other indirect questioning methods, it is generally well
received by both interviewers and respondents
\citep[e.g.,][]{Locander:1976}.  While the validation studies remain
quite rare given the difficulty of obtaining the ground truth about
sensitive behavior and attitudes, a recent study by
\citet{Rosenfeld:2014} reports that the randomized response method
recovers the truth well when compared to other methods \citep[but
see][for less favorable evidence]{kirc:15}.

On the other hand, other scholars caution that the randomized response
procedures may confuse respondents and yield noncompliance, requiring
more experienced interviewers for successful implementation
\citep[e.g.,][]{holb:kros:10a,Coutts:2011b,Wolter:2013,Hoglinger:2014}.
To address these potential problems, many researchers explain the goal
of randomized response methods to respondents
\citep[e.g.,][]{Gingerich:2010}.  Nevertheless, \citet{Coutts:2011}
find that many respondents do not believe the randomized response
technique protects anonymity even when they completely understand the
instructions.

To further assuage concerns of respondent noncompliance with
randomized response survey instructions, we propose new robust designs
that are based on less stringent assumptions than those of the
standard designs (Section~\ref{sec:ModifiedDesigns}).  For example, we
propose a design that allows for an unknown degree of noncompliance to
instructions.  We then develop the same set of methodological tools
for these modified designs so that researchers can fit multivariate
regression models and conduct power analyses.  The new designs should
address concerns, expressed frequently by applied researchers, about
the standard randomized response techniques and hence further widen
the applicability of the methodology.

All methodologies discussed in this paper are made available through
open-source software, {\tt rr: Statistical Methods for the Randomized
  Response Technique}\if0\blind \citep{blai:zhou:imai:15}\fi, which is
freely available for download at the Comprehensive R Archive Network
(\url{http://cran.r-project.org/package=rr}).  Other related software
packages for randomized response methods include the Stata module {\tt
  rrlogit}~\citep{Jann:2011b}, the R package {\tt
  RRreg}~\citep{Heck:2014}, and the (MC)SIMEX algorithm in the R
package {\tt simex}~\citep{Lederer:2013}.  Among these packages, {\tt
  RRreg} is perhaps the most comprehensive and hence is similar to our
software though there are some differences (e.g., our estimation
strategy is based on the EM algorithm whereas {\tt RRreg} uses the
standard optimization routine).

Finally, in this paper, we assume simple random sampling and do not
explore various theoretical and practical issues that may arise when
adopting different survey sampling methods.  We also do not consider
how randomized response methods can be used together with direct
questioning.  \citet{Chaudhuri:2011} explore these and other issues.

\section{Basic Designs with Known Probability}
\label{sec:Designs}

In this section, we summarize the basic designs of the randomized response
technique that have been proposed in the literature.  We classify
these designs into four types: {\it mirrored question}, {\it forced
  response}, {\it disguised response}, and {\it unrelated question}.
For each type, we provide a brief explanation, an example, and a
discussion about identification.  All four designs make two
assumptions: (1) the randomization distribution is known to
researchers, and (2) respondents comply with the instructions and
answer the sensitive question truthfully when prompted.  For some
randomized response methods, randomization is not explicitly done by
the respondent using an instrument such as coin flip.  Instead, they
may exploit a random variation that already exists (e.g., phone number
or birthday). We refer to all of these methods as randomized response
techniques.

\subsection{Mirrored Question Design}

We begin with the classic design introduced by \citet{Warner:1965},
which we call the {\it mirrored question design} (in the literature,
this design is sometimes called ``Warner's method'').  The basic idea
is to randomize whether or not a respondent answers the sensitive item
or its inverse.  As a recent example, \citet{Gingerich:2010} uses this
design to measure corruption among public bureaucrats in Bolivia,
Brazil, and Chile. The survey interviewed 2,859 bureaucrats from 30
different institutions.  Each respondent was provided a spinner and
then instructed to whirl the device without letting the interviewers
know the outcome.  The actual instruction is reproduced here,

{\spacingset{1}
\begin{verbatim}
For each of the following questions, please spin the arrow until it has made at 
least one full rotation. If the arrow lands on region A for a particular 
question, respond true or false in the space indicated only with respect to 
statement A. If the arrow lands on region B for that question, respond true or 
false in the space indicated only with respect to statement B.  Do not make any 
marks to indicate in which region the arrow fell for each question. Please 
remember that if you respond false to a statement in its negative form that 
means that the positive form of the statement is true.
\end{verbatim}}

\noindent If the spinner landed on region A, the respondent answers the following question.
{\spacingset{1}
\begin{verbatim}
A. I have never used, not even once, the resources of my institution for the 
   benefit of a political party.
\end{verbatim}
}

\noindent If the spinner landed on region B, the respondent answers its inverse.
{\spacingset{1}
\begin{verbatim}
B. I have used, at least once, the resources of my institution for the benefit of
   a political party. 
\end{verbatim}
}

Another example of this mirrored design is an ecological study to
examine whether members of marine clubs in Australia collected shells
without permits from the protected Great Barrier Reef
\citep{Chaloupka:1985}. Other applications include whether respondents
are in favor of capital punishment \citep{Lensvelt:2005b} and
legalizing marijuana use \citep{Himmelfarb:2008}.

It is straightforward to see that the response probability for the
sensitive question is identified.  Let $Z_i$ be the latent binary
response to the sensitive question for respondent $i$ (i.e., the first
statement in the aforementioned example).  We use $p$ to denote the
probability, determined by a randomization device such as a spinner,
that respondents are supposed to answer the sensitive question in the
original (rather than mirrored) format. Finally, the observed binary
response is denoted by $Y_i$.  The key relationship among these
variables is given by the following equation,
\begin{eqnarray}
  \Pr(Y_i = 1) & = & p \Pr(Z_i = 1)  + (1-p) \Pr(Z_i = 0). \label{eq:mirror}
\end{eqnarray}
Solving for $\Pr(Z_i = 1)$ yields,
\begin{eqnarray}
  \Pr(Z_i = 1) & = & \frac{1}{2p - 1} \l\{\Pr(Y_i = 1) + p - 1 \r\}. \label{eq:mirror-solve}
\end{eqnarray}
Thus, so long as $p$ is not equal to $1/2$, the response distribution
to the sensitive question is identified. 

As an extension of this design, \citet{Mangat:1990} and
\citet{Mangat:1994} propose a two-stage procedure to improve
efficiency while preserving the computational ease of the
estimator. It asks respondents who actually possess the sensitive
trait to answer truthfully. Respondents who do not have the sensitive
attribute are instructed to use the randomization device to determine
which of the mirrored questions they must answer. Thus, all `no'
answers are true negatives and only the `yes' answers are distorted
(or vice versa). Under this alternative design, noncompliance among
respondents with the sensitive trait may be higher because the privacy
protection of respondents with the sensitive attribute is completely
dependent on the cooperation of the other set of respondents who do
not possess the trait \citep{Lensvelt:2005a}.

\subsection{Forced Response Design}
\label{sec:design:forced}

We next consider the {\it forced response} design, which was first
introduced by \citet{Boruch:1971}.  Here, we describe a simpler
version of this design as introduced by \citet{Fox:1986}.  Under this
design, randomization determines whether a respondent truthfully
answers the sensitive question or simply replies with a forced answer,
`yes' or `no.'  For example, in a study on the prevalence of civilian
cooperation with militant groups in southeastern Nigeria, six-sided
dice commonly used for games in the region serve as the
randomizing device~\citep{Blair:2014}.  The survey interviewed 2,450
civilians in villages affected by militant violence.  The instructions
are reproduced here,

{\spacingset{1}
\begin{verbatim}
For this question, I want you to answer yes or no.  But I want you to consider the 
number of your dice throw.  If 1 shows on the dice, tell me no.  If 6 shows, tell 
me yes.  But if another number, like 2 or 3 or 4 or 5 shows, tell me your own 
opinion about the question that I will ask you after you throw the dice. [ TURN 
AWAY FROM THE RESPONDENT ]  Now you throw the dice so that I cannot see what comes
out.  Please do not forget the number that comes out.  [ WAIT TO TURN AROUND UNTIL
RESPONDENT SAYS YES TO: ]  Have you thrown the dice?  Have you picked it up?
\end{verbatim}
}

\noindent Thus, when the respondent rolls a one, they are forced to
respond `no' to the question; when respondents roll a six, they are
forced respond `yes.'  Finally, when respondents roll two, three,
four, or five, they are instructed to truthfully answer the following
sensitive question.

{\spacingset{1}
\begin{verbatim}
Now, during the height of the conflict in 2007 and 2008, did you know any
militants, like a family member, a friend, or someone you talked to on a
regular basis. Please, before you answer, take note of the number 
you rolled on the dice.
\end{verbatim}
}

The idea behind the forced response design is straightforward.
Because a certain proportion of respondents are expected to respond
`yes' or `no' regardless of their truthful response to the sensitive
question, the design protects the anonymity of respondents' answers.
That is, interviewers and researchers can never tell whether observed
responses are in reply to the sensitive question.

As before, let $Z_i$ represent the latent binary response to the
sensitive question for respondent $i$ and $Y_i$ represents the
observed response (1 for `yes' and 0 for `no').  Suppose further that
we use $R_i$ to represent the latent random variable, taking one of
the three possible values; $R_i = 1$ ($R_i=-1$) indicating that
respondent $i$ is forced to answer `yes' (`no'), and $R_i = 0$
indicating that the respondent is providing a truthful answer $Z_i$.
Then, the forced design implies the following equality,
\begin{eqnarray}
  \Pr(Y_i = 1) & = & p_1 + (1-p_1-p_0) \Pr(Z_i = 1) \label{eq:forced:y}
\end{eqnarray}
where $p_0 = \Pr(R_i = -1)$ and $p_1 = \Pr(R_i = 1)$.  This allows us
to derive the probability that a respondent truthfully answers `yes'
to the sensitive question,
\begin{eqnarray}
  \Pr(Z_i = 1) & = & \frac{\Pr(Y_i = 1) - p_1}{1-p_1-p_0}. \label{eq:forced:z}
\end{eqnarray}

This design is the most popular among applied researchers, and
numerous examples are found in various disciplines and methodological
illustrations.  They include a study of xenophobia and anti-Semitism
in Germany \citep{Krumpal:2012}, fabrication in job applications
\citep{Donovan:2003}, employee theft \citep{Wimbush:1997}, social
security fraud \citep{Heijden:1996}, sexual behavior and orientation
\citep{Fidler:1977}, vote choice regarding a Mississippi abortion
referendum \citep{Rosenfeld:2014}, illegal poaching among South
African farmers \citep{StJohn:2012}, use of performance enhancing
drugs \citep{Stubbe:2014}, and violation of regulatory laws by
commercial firms \citep{Elffers:2003}.  Furthermore,
\citet{Dejong:2010} expands this design to allow for ordinal responses
(e.g. a Likert scale), which they use to measure the frequency of
respondent consumer use of adult entertainment.


\subsection{Disguised Response Design}

The next design we consider is the {\it disguised response design},
which was originally proposed by \citet{Kuk:1990} (in the literature,
this design is sometimes called ``Kuk's design'').  This design was
created in order to address the problem that under the other
randomized response designs some respondents may still feel
uncomfortable providing a particular response (e.g., answering `yes')
even when interviewers do not know whether they are answering the
sensitive question. For example, \citet{Edgell:1982} use the forced
response design to study college students' experiences with
homosexuality. By fixing the outcome of the randomization device
unbeknownst to the respondents, the researchers found that 25\% of the
respondents who were forced to reply `yes' by design did not do
so. Considering this unsuccessful application of randomized response,
\citet{Heijden:1996} suggest that a disguised response design would
have been better suited given respondents had difficulties even giving
a false `yes' response.

Under the disguised response design, `yes' and `no' are replaced with
more innocuous words. This design is best understood with an example.
\citet{Heijden:2000} use the design to study fraud and malingering by
employees regarding social welfare provisions in the Netherlands
\citep[see also][]{Cruyff:2007}.  The randomization device consists of
two stacks of cards with both black and red cards.  In the right or
`yes' stack the proportion of red cards is $p=0.8$ whereas in the left
or `no' stack $1-p=0.2$.  Respondents are asked to draw one card from
each stack. Instead of answering `yes' (`no') to a sensitive question,
they are instructed to name the color of the card from the right
(left) stack.  The original instruction reads as follows,

{\spacingset{1}
\begin{verbatim}
I have two stacks of cards and a box behind in which I place the cards. 
[GIVE THE BOX TO THE RESPONDENT AND LOOK AT IT TOGETHER.] In the box, you find a 
card on which it is written what the stack means: the right-hand stack is the 
`yes' stack, and the left-hand stack is the `no' stack. [LET INTERVIEWEE LOOK AND 
GIVE DIRECTIONS WITH THE NEXT EXPLANATION.] In the `yes' stack [POINT TO THE 
RIGHT-HAND STACK] there are more red cards than in the `no' stack [POINT TO THE 
LEFT-HAND STACK, RESPONDENT MAY CHECK]. If you want, you may shuffle the two 
stacks [SEPARATELY]. Now, please take from each stack an arbitrary card. You may 
take the card on top or from within the stack. [TAKE A CARD FROM EACH STACK]
Nobody but you can see the colors of your cards; when you mention a card color, 
we do not know the stack from which you took the card. Thus, your privacy is 
guaranteed: your answer will always remain a secret. [...] I propose that we now 
try out a few questions.
\end{verbatim}
}

\noindent Then, respondents answer with `red' or `black' to the set of
questions, which include the following:

{\spacingset{1}
\begin{verbatim}
At a social services check-up, have you ever acted as if you were sicker or 
less able to work than you actually are?

Have you ever noticed an improvement in the symptoms causing your disability, 
for example in your present job, in volunteer work, or the chores you do at home, 
without informing the Department of Social Services of this change?
\end{verbatim}
}

The identification strategy for the probability of answering `yes' to
a sensitive question is exactly the same as that for the mirrored
response design.  Let $Z_i$ represent the latent response to the
sensitive question with $Z_i = 1$ ($Z_i = 0$) indicating an
affirmative (negative) answer.  We use $p$ to represent the proportion
of red cards in the right or `yes' stack ($1-p$ the proportion of red
cards in the left or `no' stack).  Finally, let $Y_i$ denote the
observed binary response where $Y_i = 1$ ($Y_i = 0$) represents the
reply `red' (`black').  Then, the key relationship between the
probability of observing the answer `red' and the probability of
affirmative response towards the sensitive item is described by
equation~\eqref{eq:mirror}, and therefore the latter quantity is given
by equation~\eqref{eq:mirror-solve}.

\subsection{Unrelated Question Design}
\label{subsec:unrelated}

The final design we consider is the {\it unrelated question design},
which was developed by \citet{Greenberg:1969,Greenberg:1971}.  Under
this design, randomization determines whether a respondent should
answer a sensitive question or an unrelated, non-sensitive question.
Unlike the other designs, this design introduces an unrelated question
in order to increase respondents' compliance with survey
instruction. Furthermore, \citet{Moors:1971} shows that this design is
more efficient than the mirror question, and it allows for
quantitative responses.

For example, \citet{Chi:1972} apply the unrelated question design to
study the incidence of induced abortions in Taiwan using pieces from
the regionally popular game \textit{Go}. The researchers interviewed a
random sample of 2,497 women between ages 15 and 49. Census data was
used to estimate the proportion of the unrelated, innocuous question
about the respondent's year of birth.  We reproduce the instructions
here,

{\spacingset{1}
\begin{verbatim}
Here is a bag; in it there are stones from the game `Go,' some colored black and 
others white.  Please take one stone out, and see by yourself what color it is, 
black or white.  Don't let me know whether it is black or white, but be sure 
you know which it is. If you take a black one, answer the question: "Have you 
ever had an induced abortion?"  If you take a white one, answer the question: 
"Were you born in the lunar year of the horse?" 
\end{verbatim}
}

\noindent Similar studies on abortion rates have been conducted in
North Carolina \citep{Abernathy:1970}, Mexico
\citep{Lara:2004,Lara:2006}, and Turkey \citep{Tezcan:1981}.  Other
applications of the unrelated question include a criminology study of
self-reported arrests in Philadelphia \citep{Tracy:1981}, a
sociological assessment concerning the concealment of deaths in the
household from local authorities in the Philippines
\citep{Madigan:1976}, and self-reported failure of classes by college
students \citep{Lamb:1978}.

Let $p$ denote the probability that respondents receive the sensitive
question.  This probability is assumed to be known.  In the above
example, it equals the proportion of black stones in the bag.  
We use $Z_i$ to denote the binary latent response to the
sensitive question with $Z_i = 1$ ($Z_i = 0$) representing the
affirmative (negative) answer.  Furthermore, let $q$ represent the
probability of answering `yes' to the unrelated question.  It is
assumed that researchers also know this probability: in the
aforementioned example, the census data are used to determine it.
Then, if we use $Y_i$ to denote the observed binary response, the key
estimating equation is given by,
\begin{eqnarray}
  \Pr(Y_i = 1) & = & p \Pr(Z_i = 1) + (1-p)q. \label{eq:unrelated}
\end{eqnarray}
This yields the identification of the response distribution to the
sensitive question,
\begin{eqnarray*}
  \Pr(Z_i = 1) & = & \frac{1}{p} \l\{\Pr(Y_i = 1) - (1-p) q \r\}.
\end{eqnarray*}

As variants of this design, \citet{Yu:2008} and \citet{Tan:2009}
propose two designs that do not require a randomizing device: the
triangular and crosswise designs. Both designs make use of an
unrelated, non-sensitive question (e.g. whether the respondent is born
between August and December) that is assumed to be independent of the
sensitive item (e.g. whether the respondent is a drug user). The
triangular design asks respondents to mark one of two statements: (1)
neither characteristics are true or (2) at least one of the
characteristics is true. Relying on the same setup, the crosswise
design asks respondents to choose one of the following statements: (1)
both or neither characteristics are true or (2) one of the
characteristics is true.  \citet{Gingerich:2014} use the crosswise
design to develop a joint model that combines indirect and direct
questioning within the same survey in order to determine whether a
topic is sufficiently sensitive to justify indirect questioning.
Other works that study these designs include \citet{Coutts:2011b},
\citet{Jann:2011}, \citet{Hoglinger:2014}, and
\citet{Korndorfer:2014}.

\section{Statistical Analysis of the Basic Designs}
\label{sec:Statistics}

In this section, we describe how to analyze data from the randomized
response method under the four basic designs reviewed in the previous
section.  We begin by presenting the likelihood framework for
conducting a multivariate regression analysis, an essential tool for
researchers who wish to understand the respondent characteristics that
are associated with the sensitive attitudes and behavior under
investigation.  Within this framework, researchers can then generate
predicted probabilities for the sensitive item given characteristics.
We also show how to use the sensitive attitudes and behavior inferred
from the multivariate regression analysis as a predictor for an
outcome regression model.  Finally, we demonstrate how to conduct
power analysis for the four basic designs of randomized response
method.

\subsection{Multivariate Regression Model}
\label{subsec:model}

\begin{table}[!t]
  \centering 
  \spacingset{1}
  \begin{tabular}{p{1.5in} p{3.5in} p{1in}}
    \hline\hline 
    & & {\bf Model} \\
    {\bf Design} & {\bf Design Parameters} & {\bf Parameters} \\\hline 
    \multirow{2}{*}{Mirrored Question}
    & $p$: probability of receiving the sensitive & $c = 2p - 1$ \\ 
    &  question in its original format as opposed to its inverse & $d = 1- p$ \\\hline
    \multirow{3}{*}{Forced Response}
    & $p$: probability of answering truthfully  & $c = p$ \\ 
    & $p_1$: probability of forced `yes' & $d = p_1$ \\
    & $p_0$: probability of forced `no' ($p_0 = 1-p-p_1$) & \\\hline
    \multirow{2}{*}{Disguised Response}
    & $p$: probability of selecting a red card from the  & $c = 2p - 1$ \\ 
    & `yes' stack & $d = 1 - p$ \\ \hline
    \multirow{4}{*}{Unrelated Question}
    & $p$: probability of receiving the sensitive & $c = p$ \\ 
    &  question as opposed to the unrelated question & $d = (1 - p)q$ \\
    & $q$: probability of answering `yes' to the unrelated question
    (assumed to be independent of covariates, otherwise needs to be modeled) & \\\hline \hline
  \end{tabular}
  \caption{Correspondence between Design and Model Parameters. The table
    shows, for each design of randomized response method,
    the correspondence between design and model parameters.  The general
    model, which can be applied to all four designs, is given in the
    likelihood function of equation~\eqref{eq:likelihood}.}
\label{tab:parameters}
\end{table}


The goal of multivariate regression analysis is to characterize how a
vector of respondent characteristics $X_i$ is associated with the
latent response to the sensitive question $Z_i$.  We define this
regression model as,
\begin{eqnarray}
  \Pr(Z_i = 1 \mid X_i = x) & = & f_\beta(x) \label{eq:model}
\end{eqnarray}
where $\beta$ is a vector of unknown parameters.  A popular choice of
the parametric model is the logistic regression, $f_\beta(x) =
\exp(x^\top \beta)/\{1+\exp(x^\top \beta)\}$.  Using
equations~\eqref{eq:mirror},~\eqref{eq:forced:y},~and~\eqref{eq:unrelated},
we can construct the likelihood function as,
\begin{eqnarray}
\cL(\beta \mid \{X_i, Y_i\}_{i=1}^n) & = & \prod_{i=1}^{N} \lbrace c
 f_\beta(X_i) + d\rbrace^{Y_i} \lbrace 1 - (c 
f_\beta(X_i) + d) \rbrace^{1 - Y_i} \label{eq:likelihood}
\end{eqnarray}
where $c$ and $d$ are known constants determined by each of the four
basic designs.  For example, under the mirrored question design,
$c=2p-1$ and $d=1-p$ where $p$ is the probability of answering the
sensitive question in the original format.  Table~\ref{tab:parameters}
summarizes the relationship between the model parameters $(c,d)$ and
the design parameters under each of the four basic designs.

Our contribution here is to point out that all four designs can be
analyzed under the single likelihood function given in
equation~\eqref{eq:likelihood}.  In the literature, \citet{Hout:2007}
show that the same likelihood function applies to the forced response
and mirrored response designs \citep[see also][]{Scheers:1988}.
\citet{Heijden:1996} develop a similar likelihood framework for the
forced response and disguised response designs.  Additionally,
\citet{Warner:1965} considers the linear regression model while
\citet{Winkler:1979} and \citet{Ohagan:1987} explore its Bayesian
extensions.

It is important to emphasize that an additional assumption is made
when applying the likelihood function in
equation~\eqref{eq:likelihood} to the unrelated question design.
Specifically, it is assumed that the response to the unrelated
question $q$ is independent of the covariates $X_i$.  In the case of
the empirical application discussed in Section~\ref{subsec:unrelated},
whether a respondent is born in a certain lunar year is assumed to be
independent of whatever covariates that will be included in the model
$f_\beta(x)$.  If this assumption is relaxed, then the design
parameter $q$ must be modeled as a function of covariates as
$q_\gamma(X_i)$ where $\gamma$ is a vector of unknown parameters.
This in turn implies that the model parameter $d$ needs to be a
function of $X_i$.  The likelihood function for the unrelated question
design, then, becomes,
\begin{equation}
\cL(\beta, \gamma \mid \{X_i, Y_i\}_{i=1}^n) \ = \ \prod_{i=1}^{N} \{ p
 f_\beta(X_i) + (1-p)q_\gamma(X_i)\}^{Y_i} \l[ 1 - \{p 
f_\beta(X_i) + (1-p)q_\gamma(X_i)\}\r]^{1 - Y_i}. \label{eq:likelihood-unrelated}
\end{equation}
To avoid this unnecessary modeling assumption about responses to the
unrelated question, researchers should choose an unrelated question
whose responses are known to be independent of respondent
characteristics. 

One approach, which guarantees that the required independence
assumption is met, is to employ a two-stage randomization process. For
instance, in addition to the first randomization device, which
determines whether the respondent answers the sensitive question or
the unrelated question (e.g., drawing from the bag of black and white
stones in \citet{Chi:1972}), respondents are instructed to flip a
coin.  Upon selecting a white stone, the respondent is prompted to
answer the following unrelated question, `Did you flip heads?' While
this process waives the need to model $q$, it also adds a layer of
complexity to the design procedure.


\subsection{Estimation}
\label{subsec:estimation}

\citet{Hout:2007} focuses on the generalized linear model framework
and utilizes iteratively reweighted least squares.  For example, if we
assume the logistic regression for $f_\beta(X_i)$, we have,
\begin{eqnarray*}
  \mu_i & \equiv & c f_\beta(X_i) + d \quad {\rm and} \quad g(\mu_i) \ = \ \log \frac{\mu_i - d}{c+d-\mu_i} \ = \ \beta X_i
\end{eqnarray*}
where $g(\cdot)$ is a monotonic and differentiable link function with
its domain equal to $(d,c+d)$.  Then, the standard generalized linear
model (GLM) routine can be used to obtain the maximum likelihood
estimate of $\beta$.

As an alternative and more generally applicable estimation method, we
develop the Expectation-Maximization (EM) algorithm below to maximize
the likelihood function in equation~\eqref{eq:likelihood}
\citep{demp:lair:rubi:77}.  The advantage of the proposed algorithm is
that it only requires the estimation routine for the underlying model
$f_\beta(X_i)$ and hence is applicable to a wide range of models
beyond the GLMs.  While we develop a separate EM algorithm for each
design, they all maximize the same observed-data likelihood function
given in equation~\eqref{eq:likelihood}.

\paragraph{Mirrored Question Design.} We first develop the EM
algorithm under the mirrored question design.  Let the latent
indicator variable $T_i = 1$ ($T_i = 0$) denote the scenario where
respondent~$i$ answers the sensitive question in the original
(mirrored) format.  Under this design, the complete-data likelihood
function is given as follows,
\begin{eqnarray*} 
  \cL_{com} (\beta \mid \{Y_i, T_i, X_i\}_{i=1}^n) 
%& = &
%  \prod_{i=1}^N \{ f_\beta(X_i)^{Z_i} (1-f_\beta(X_i))^{1-Z_i}
%  \}^{T_i} \{ f_\beta(X_i)^{1-Z_i} (1-f_\beta(X_i))^{Z_i} \}^{1-T_i} \\
%  & = & \prod_{i=1}^N f_\beta(X_i)^{T_i Z_i + (1-T_i)(1-Z_i)} 
%  \{1-f_\beta(X_i)\}^{T_i(1-Z_i) + (1-T_i)Z_i}
  & = & \prod_{i=1}^N f_\beta(X_i)^{T_i Y_i + (1-T_i)(1-Y_i)}
  \{1-f_\beta(X_i)\}^{T_i(1-Y_i) + (1-T_i)Y_i}. 
\end{eqnarray*}
The E-step consists of calculating the following conditional
expectations,
\begin{equation*}
  \E(T_i \mid X_i = x, Y_i = y) 
%   & = & \frac{\Pr(Y_i = y \mid T_i = 1, X_i =
%     x) \Pr(T_i =
%   1)}{\sum_{t=0}^1 \Pr(Y_i = y \mid Z_i = z,
%  T_i = t,  X_i = x) \Pr(T_i = t)}\nonumber \\
  \ = \ \frac{p f_\beta(x)^y (1-f_\beta(x))^{1-y}}{p f_\beta(x)^y (1-f_\beta(x))^{1-y} + (1-p)  f_\beta(x)^{1-y} (1-f_\beta(x))^y}.  
\end{equation*}
Then, the M-step maximizes the following objective function with
respect to $\beta$,
\begin{eqnarray*}
  \sum_{i=1}^n \{1 - Y_i - (1-2Y_i)w_T(X_i,Y_i)\}
  \log f_\beta(X_i) 
+ \{Y_i + (1-2Y_i) w_T(X_i,Y_i) \} \log (1-f_\beta(X_i))
\end{eqnarray*}
where $w_T(X_i,Y_i) = \E(T_i \mid X_i, Y_i)$.  Given the starting
values for $\beta$, the algorithm proceeds by alternating the E-step
(using the values of $\beta$ from the previous iteration) and the
M-step.  In particular, the M-step can be implemented via a weighted
regression fitting routine for $f_\beta(x)$.

Finally, we can use the following equation to calculate the posterior
prediction of latent responses to the sensitive question for each
respondent in the sample, i.e., 
\begin{equation}
  \Pr(Z_i = 1 \mid X_i = x, Y_i = y) 
% & = & \frac{\sum_{t=0}^1 \Pr(Y_i = y \mid Z_i = 1,
%  T_i = t,  X_i = x) \Pr(Z_i = 1 \mid X_i = x) \Pr(T_i = t)}{\sum_{z=0}^1 \sum_{t=0}^1 \Pr(Y_i =% y \mid Z_i = z,
%  T_i = t,  X_i = x) \Pr(Z_i = z\mid X_i = x) \Pr(T_i = t)} \nonumber \\
\ = \ \frac{p^y(1-p)^{1-y} f_\beta(x) }{p^y(1-p)^{1-y} f_\beta(x) + p^{1-y}(1-p)^y (1-f_\beta(x))}. \label{eq:EZpost.mirror}
\end{equation}

\paragraph{Forced Response Design.} Next we consider the forced
response design.  Let $R_i$ denote the latent randomization variable
where $R_i = 1$ ($R_i = -1$) indicates that respondents are forced to
answer `yes' (`no') and $R_i = 0$ implies that the respondent answers
the sensitive question truthfully.  Then, the complete-data likelihood
function is given by,
\begin{equation}
  \cL_{com} (\beta \mid \{X_i, Y_i, R_i\}_{i=1}^n) 
\ \propto \
%  \prod_{i=1}^n \{f_\beta(X_i)^{Z_i}
%  (1-f_\beta(X_i))^{1-Z_i}\}^{\mathbf{1}\{R_i = 0\}} \\
%& = &
  \prod_{i=1}^n \{f_\beta(X_i)^{Y_i}
  (1-f_\beta(X_i))^{1-Y_i}\}^{\mathbf{1}\{R_i = 0\}} \label{eq:llik:forced}
\end{equation}
where $Y_i=Z_i$ when $R_i=0$ and the likelihood function is constant
in $\beta$ when $R_i \ne 0$.

The E-step is given by the following conditional expectations,
\begin{equation*}
\E(\mathbf{1}\{R_i = 0\} \mid X_i = x, Y_i = y) 
%& = &\frac{\Pr(Y_i =
%  y \mid R_i = 0, X_i = x)\Pr(R_i = 0)}{ \sum_{r=-1}^1 \Pr(Y_i =
%  y \mid R_i = r, Y_i = y)
%\Pr(R_i = r)} \\
\ = \ \frac{p f_\beta(x)^y (1-f_\beta(x))^{1-y}}{p f_\beta(x)^y (1-f_\beta(x))^{1-y} + p_1^y p_0^{1-y} }. 
\end{equation*}
Then, the objective function for the M-step is,
\begin{equation*}
  \sum_{i=1}^n w_{R}(X_i, Y_i) \{Y_i \log f_\beta(X_i) + (1-Y_i) \log (1-f_\beta(X_i))\}
\end{equation*}
where $w_{R}(X_i,Y_i) = \E(\mathbf{1}\{R_i = 0\} \mid X_i, Y_i)$. The
algorithm iterates between the E and M steps where the latter is
carried out by fitting the weighted regression model.

Finally, we use the following conditional expectation to calculate the
posterior prediction of responses to the sensitive question for each
respondent in the sample,
\begin{equation}
  \Pr(Z_i = 1 \mid X_i = x, Y_i = y) 
%  & = &  \frac{\sum_{r=-1}^1 \Pr(Y_i =
%  y \mid R_i = r, Z_i = 1, X_i = x)\Pr(Z_i = 1 \mid X_i =
%  x)\Pr(R_i = r)}{\sum_{z=0}^1 \sum_{r=-1}^1 \Pr(Y_i =
%  y \mid R_i = r, Z_i = z, Y_i = y)\Pr(Z_i = z \mid X_i =
%  x)\Pr(R_i = r)} \\
\ = \ \frac{(p+p_1)^y p_0^{1-y} f_\beta(x)}{p
  f_\beta(x)^y (1-f_\beta(x))^{1-y} + p_1^y  p_0^{1-y} }. \label{eq:EZpost.forced}
\end{equation}

\paragraph{Disguised Response Design.} For the disguised response
design, the latent response to the sensitive item, $Z_i$, determines
whether a respondent draws a card from the `yes' stack ($Z_i = 1$) or
`no' stack ($Z_i = 0$).  In each stack, the probability of drawing a
`red' card ($Y_i = 1$) is determined by $p$ and $1-p$ for the `yes'
and `no' stacks, respectively.  Thus, the complete-data likelihood
function is given by,
\begin{equation*}
  \cL_{com}(\beta \mid \{X_i, Y_i, Z_i\}_{i=1}^n \} \ = \ \prod_{i=1}^n
  \{f_\beta(X_i) p^{Y_i} (1-p)^{1-Y_i}\}^{Z_i}
  \{(1-f_\beta(X_i))p^{1-Y_i} (1-p)^{Y_i}\}^{1-Z_i}.   
\end{equation*}
Then, the E-step of the EM algorithm is given by,
\begin{eqnarray*}
  E(Z_i \mid X_i = x, Y_i = y) & = & \frac{f_\beta(X_i) p^{y}
    (1-p)^{1-y}}{f_\beta(X_i) p^{y} (1-p)^{1-y} +
    (1-f_\beta(X_i))(1-p)^y p^{1-y}}
\end{eqnarray*}
which also gives the posterior prediction of response to the sensitive
question.  Finally, the objective function for the M-step is given by,
\begin{equation*}
  \sum_{i=1}^n w_Z(X_i,Y_i) \log f_\beta(X_i) + (1-w_Z(X_i,Y_i)) \log (1-f_\beta(X_i))
\end{equation*}
where $w_Z(X_i,Y_i) = \E(Z_i \mid X_i, Y_i)$. 

\paragraph{Unrelated Question Design.}  Finally, we develop an EM
algorithm for the unrelated question design.  \citet{Bourke:1988}
propose the EM algorithm for estimating the population proportion of
affirmatively answering the sensitive question.  Here, we generalize
their algorithm for multivariate regression analysis.  Let $S_i$
denote the latent binary variable, which indicates whether
respondent~$i$ answers the sensitive item ($S_i = 1$) or the unrelated
question ($S_i = 0$).  Then, the complete-data likelihood function is
given by,
\begin{equation*}
  \cL_{com}(\beta, \gamma \mid \{X_i, Y_i, S_i\}_{i=1}^n) \ = \ \prod_{i=1}^n
  \{f_\beta(X_i)^{Y_i} (1-f_\beta(X_i))^{1-Y_i}\}^{S_i} \{q_\gamma(X_i)^{Y_i} (1-q_\gamma(X_i))^{1-Y_i}\}^{1-S_i}.
\end{equation*}

The E-step is given by the following conditional expectations,
\begin{equation*}
  \E(S_i \mid X_i = x, Y_i = y) 
%  & = &  \frac{\Pr(Y_i = y \mid X_i = x, S_i = 1)\Pr(S_i = 1)}{\sum_{s=0}^1  
%    \Pr(Y_i = y \mid X_i = x, S_i = s)\Pr(S_i = s)} \nonumber\\
  \ = \ \frac{p f_\beta(x)^y (1-f_\beta(x))^{1-y}}{p f_\beta(x)^y (1-f_\beta(x))^{1-y} +
    (1-p)q_\gamma(x)^y (1-q_\gamma(x))^{1-y}}.
\end{equation*}
Given this E-step, the M-step maximizes the following objective
function,
\begin{eqnarray*}
  & & \sum_{i=1}^n w_{S}(X_i,Y_i) \{Y_i \log f_\beta(X_i) + (1-Y_i) \log
  (1-f_\beta(X_i))\} \nonumber \\
 & & \hspace{.5in} + (1- w_{S}(X_i,Y_i)) \{Y_i \log q_\gamma(X_i) + (1-Y_i) \log
  (1-q_\gamma(X_i))\}
\end{eqnarray*}
where $w_S(X_i,Y_i) = \E(S_i \mid X_i = x, Y_i = y)$.  This step is
done by fitting the weighted regression models for $f_\beta(X_i)$ and
$q_\gamma(X_i)$, separately.

Finally, under the unrelated question design, the posterior prediction
of responses to the sensitive question cannot be calculated unless one
models the association between responses to the sensitive question and
those to the unrelated question, conditional on the respondent
characteristics $X_i$.  On the other hand, if we assume the
conditional independence between them given $X_i$, then the posterior
probability is given by,
\begin{eqnarray*}
  \Pr(Z_i = 1 \mid X_i = x, Y_i = y) 
%  & = & \frac{
%    \sum_{s=0}^1 \Pr(Y_i = y \mid X_i = x, Z_i = 1, S_i = s)\Pr(Z_i = 1 \mid X_i =
%    x)\Pr(S_i = s)}{\sum_{s=0}^1 
%    \Pr(Y_i = y \mid X_i = x, S_i = s)\Pr(S_i = s)} \nonumber \\
  & = & \frac{\{py + (1-p)q_\gamma(x)^y(1-q_\gamma(x))^{1-y}\} f_\beta(x)}{p 
     f_\beta(x)^y (1-f_\beta(x))^{1-y} + (1-p)q_\gamma(x)^y (1-q_\gamma(x))^{1-y}}.
\end{eqnarray*}


\subsection{Using Randomized Response as a Predictor}
\label{subsec:predict}

In many cases, researchers wish to use randomized response as a
predictor in an outcome regression.  \citet{imai:park:gree:14} develop
such a method for the item count technique (or list experiment).
Here, we apply the same modeling strategy to the randomized response
methods.  To illustrate, we consider the forced response design
although the same idea can be applied to the other designs.  Let $V_i$
represent the outcome variable of interest, and suppose that
researchers are interested in fitting the following outcome regression
model, $g_\theta(V_i \mid X_i, Z_i)$ where $\theta$ is comprised of
the parameters of the outcome model.  For example, if the outcome
model is the normal linear regression, we have $g_\theta(V_i \mid X_i,
Z_i) = \mathcal{N}(\alpha + \gamma^\top X_i + \delta Z_i, \sigma^2)$
where $\theta=(\alpha,\gamma,\delta,\sigma^2)$, $Z_i$ is the latent
randomized response variable, and the coefficient of interest is
$\delta$.

Since $Z_i$ is not directly observed, we develop an EM algorithm to
fit this model.  We begin by assuming that $V_i$ and $Y_i$ are
conditionally independent given $X_i$.  This assumption can be relaxed
by modeling their joint distribution, but here we maintain this
assumption for the sake of simplicity.  Then, the (observed-data)
likelihood function for the combined model is given by,
\begin{eqnarray}
 \cL(\theta, \beta \mid \{V_i, X_i, Y_i \}_{i=1}^n) 
& = & \prod_{i=1}^{N} \l\{f_\beta(X_i) g_\theta(V_i \mid X_i,
1)(1-p_0)^{Y_i} p_0^{1-Y_i} + \nonumber \r. \\ & & \l. (1-f_\beta(X_i)) g_\theta(V_i \mid X_i, 0)p_1^{Y_i} (1-p_1)^{1-Y_i} \r\}. 
\label{eq:outcomelik}
\end{eqnarray}
With $R_i$ denoting the latent randomization variable indicating
whether respondents answer the sensitive question, the complete-data
likelihood function is given as,
\begin{equation*}
\cL_{com}(\theta, \beta \mid \{V_i, X_i, R_i, Y_i \}_{i=1}^n)
\ \propto \ \prod_{i=1}^{N} \l[\{g_\theta(V_i \mid X_i, 1) f_\beta(X_i) \}^{Y_i} \{g_\theta(V_i \mid X_i, 0)(1 - f_\beta(X_i)) \}^{1-Y_i}\r]^{\mathbf{1}\{R_i = 0\}}
\end{equation*}
where $Y_i = Z_i$ when $R_i = 0$, and the likelihood function is
constant in $\beta$ when $R_i \neq 0$.  The E-step is given by the
following conditional expectation,
\begin{eqnarray*}
& & \E(\textbf{1}\{R_i =0\} \mid X_i = x, Y_i = y, V_i = v) \nonumber\\
& = &
\frac{p g_\theta(v \mid x, y) f_\beta(x)^y (1 - f_\beta(x))^{1-y}}{p g_\theta(v \mid x, y) f_\beta(x)^y (1 - f_\beta(x))^{1-y}  + p_1^y p_0^{1-y}\{g_\theta(v \mid x, 1)f_\beta(x) + g_\theta(v \mid x, 0)(1 - f_\beta(x))\} }. 
\end{eqnarray*}
Finally, the M-step maximizes the following complete-data
log-likelihood function,
\begin{equation*}
\sum_{i=1}^{n} w_R(X_i, Y_i, V_i) \cdot [Y_i \{\log f_\beta(X_i) + \log g_\theta(V_i \mid X_i, 1) \} 
+ (1 -Y_i)\{ \log(1 - f_\beta(X_i)) + \log g_\theta(V_i \mid X_i, 0) \}]
\end{equation*}
where $w_R(X_i, Y_i, V_i) = \E(\mathbf{1}\{R_i =0\} \mid X_i, Y_i,
V_i)$.

\subsection{Power Analysis}
\label{subsec:basicpower}

When choosing among the aforementioned four basic designs and
determining the model parameters under each design, one important
consideration is statistical efficiency.  Here, we show how to conduct
power analysis under each design.  The literature appears to contain
surprisingly few results about efficiency and power analysis.  The
only relevant work we find is \citet{Lakshmi:1992} who derive a power
function to test the probability of respondent noncompliance under a
mirrored question design.  While others compare efficiency across
various designs \citep{Moors:1971, Pollock:1976, Scheers:1988,
  Umesh:1991, Lensvelt:2005a}, they fall short of providing a unified
framework for conducting power analysis to help applied researchers
design randomized response surveys.  Our analysis fills this important
gap in the literature.

Without loss of generality, we consider the likelihood function in
equation~\eqref{eq:likelihood} with no covariates, i.e.,
$f = f_\beta(1) = \exp(\beta)/\{1+\exp(\beta)\}$.  Again, this $f$ is
the probability of possessing the sensitive trait.  The unified model
introduced in Section~\ref{subsec:model} makes this analysis
straightforward.  To begin, we derive the Fisher information with
respect to $f$ under this unified model,
\begin{eqnarray}
\label{eq:basicfisher}
 \mathcal{I}(c,d,f) & = & \E\l[\l(\frac{\partial}{\partial f} \log L(f \mid
  \{Y_i\}_{i=1}^n)  \r)^2\r] 
%  & = & c^2 \l[\frac{\Pr_\beta(Y_i = 1)}{(c f_\beta + d)^2} +
%  \frac{1-\Pr_\beta(Y_i = 1)}{\{-c f_\beta + (1-d)\}^2} \r] \\
  \ = \ \frac{c^2}{(c f + d)\{1-(cf + d)\}}. 
\end{eqnarray}
In addition, the standard error of $\hat{f}$ is given by,
\begin{eqnarray}
  \sigma(c,d,f,n) & = & \frac{1}{c\sqrt{n}}\sqrt{(c f + d)\{1-(c f + d)\}}
\end{eqnarray}
where $n$ is the sample size.  For each design, we can rewrite both
the Fisher information and standard error as the function of design
parameters using the relationships between the model and design
parameters given in Table~\ref{tab:parameters}.

Finally, we derive power functions under all designs.  The power
function determines the probability that a test procedure will reject
a null hypothesis $H_0: f = f_0$ at significance level $\alpha$ when
the true value of $f$ is equal to $f^\ast$.  We first derive an
approximate power function for a one-sided hypothesis test where the
null hypothesis is $H_0: f = f_0$ and the alternative hypothesis is
either $H_1: f > f_0$ or $H_1: f < f_0$,
\begin{eqnarray}
\label{eq:powerfunc1}
  \psi(c,d,n,f_0,f^\ast,\alpha) & = & 1 -
  \Phi\l[\frac{f_0 - f^\ast + \Phi^{-1}(1 - \alpha)  \sigma(c,d,f_0,n)}{\sigma(c,d,f^\ast,n)} \r]
\end{eqnarray}
where $\Phi(\cdot)$ is the cumulative distribution function of the
standard normal distribution.  Similarly, the power function for a
two-sided hypothesis test where the alternative hypothesis is $H_1:f
\ne f_0$ is given by,
\begin{eqnarray}
\label{eq:powerfunc2}
\psi(c,d,f_0,f^\ast,n,\alpha) & = & 1 -
\Phi\l[\frac{f_0 - f^\ast  + \Phi^{-1}(1 -
  \alpha/2)  \sigma(c,d,f_0,n)}{\sigma(c,d,f^\ast,n)}
\r]  \nonumber\\ & & \hspace{.75in} + \Phi\l[\frac{f_0 - f^\ast - \Phi^{-1}(1 - \alpha/2)  \sigma(c,d,f_0,n)}{\sigma(c,d,f^\ast,n)} \r].
\end{eqnarray}

Several notable findings follow from these results. First, for a fixed
sample size, significance level, and functional form, the power for
any two designs that have identical values of $c$ and $d$ will also be
identical. This means that the mirrored question and disguised
response designs with shared design parameter $p$ will have the same
statistical power. In addition, for the forced response and unrelated
question designs, the power will be identical when they share the same
design parameter $p$ and the forced response parameter $p_1$ is equal
to $(1-p)\cdot q$ for the unrelated question design.

\begin{figure}[t]
<<label = powerComparisonFigure, fig.height = 7, fig.width = 8>>=
              
par(mfrow = c(3,3), oma = c(1.05, 1.5, 3.5, 0.25), mar = c(2, 3, 0, 0), mgp = c(2, .7, 0), tck = -.015)
              
linetype <- c("dashed", "dotted")

for(n in c(500, 1000, 2500)) {
      for(z in c(.1, .2, .3)) {
      plot(0,1, type = "n", xlim = c(0, 1), ylim = c(0, 1), axes = F, xlab = "", ylab = "")
              
              p1.vals.plot <- c(.1, .5)
              for(j in 1:length(p1.vals.plot)){
              
              lines(power.plot(n = n, p1 = p1.vals.plot[j], presp = z, design = "forced-known"), 
              lty = linetype[j], lwd = 2)

              }
              
              mirrored.plot.df <- power.plot(n = n, presp = z, design = "mirrored")
              lines(mirrored.plot.df[mirrored.plot.df[,1] < .49, ], lty = "solid", lwd = 1.5)
              lines(mirrored.plot.df[mirrored.plot.df[,1] > .51, ], lty = "solid", lwd = 1.5)
              
              if(n == 500 & z == .1){
              text(0.615, 0.18, "Mirrored and\nDisguised", cex = 0.7, pos = 4)
              text(0.4, 0.45, paste("Forced and\nUnrelated\n0.5", sep = ""), cex = 0.7, pos = 4)
              text(0.225, 0.55, paste("Forced and\nUnrelated\n0.1", sep = ""), cex = 0.7, pos = 2)
              }
              
              if(z == .2 & n == 500)
              mtext(side = 3, "Proportion with Sensitive Trait", line = 2.25)
                
              if(n == 500)
              mtext(side = 3, paste("Pr(Z = 1) =", z), cex = .8, line = .75)
              
              if(z == .1)
              mtext(side = 2, paste("N =", n), line = 3.5, cex = .8)
              
              
              axis(1, cex.axis = .8)
              axis(2, las = 1, cex.axis = .8)
              
              mtext(side = 2, "Power", line = 2, cex = 0.7)
              if(n == 2500)
              mtext(side = 1, "Value of p", cex = 0.7, line = 2)
              }
        }                   
              @
\caption{Comparison of Power across the Four Standard Designs. First, the 
power for the forced response and unrelated question design with 
$p_1 = (1-p)\cdot q = .1$ is displayed (dashed lines). Second, the power 
for these designs with $p_1 = (1-p)\cdot q = .5$ is displayed (dotted lines). 
Third, the power for the mirrored question and disguised response designs 
is displayed (solid lines).}
\label{fig:compare:p}
\end{figure}

Now we can compare the power across designs and for different 
 parameter values within each design. 
Figure~\ref{fig:compare:p} displays a comparison for four designs and
with several realistic design parameter values for each design.
There are three notable implications of these comparisons. First,
the mirrored and disguised designs have the least power when 
$p$ is close to one half (note the design cannot be used with $p=.5$).
For a sample size of 500 and a proportion of ``yes'' responses to the 
sensitive item of $.1$, for example, power reaches the typical 
threshold of $.8$ only when $p \leq .25$ or $p \geq .75$ (see
top left plot, Figure~\ref{fig:compare:p}).

Second, higher values of $p$ for both the forced response and 
unrelated question designs yield higher power to detect the 
sensitive responses. This makes sense: the higher the value of
$p$, the less noise unrelated to the sensitive item responses
that is introduced. For example, with a sample size of 1000 and
a proportion of ``yes'' responses to the sensitive item of $.1$, 
power only reaches the threshold of $.8$ when $p \geq .4$.

Third, given a choice of $p$, it is optimal to either choose 
small (large) values or large (small) values of $p_1$ ($p_0$). 
That is, the further $p_1$ and $p_0$ are 
from $.5$ in either direction, the higher the power. 
Figure~\ref{fig:compare:p1} in the Appendix displays the power 
for the forced design
with varying values of $p_1$. For any value of $p$, the higher
the $p_1$ the lower the power until $.5$, when the relationship
reverses. 
These findings also yield design advice for the 
unrelated question design. Since $p_1 = (1-p) \cdot q$, we 
also know that values of $(1-p)\cdot q$ closer to 0 and 1 are 
preferred to values closer to $.5$. For example, for a study with
a sample size of 2500, a proportion of `yes' responses to the
sensitive item of $.1$, and $p$ set to $.2$, power only reaches
the standard threshold of $.8$ when $p_1 < .2$ or when 
$p_1 = .8$ (see bottom left plot, Figure~\ref{fig:compare:p1} in 
the Appendix).


\subsection{Comparison of the Basic Designs}
\label{subsec:Comparison}

\begin{table}[!t]
\centering 
\spacingset{1}
\begin{tabular}{>{\raggedright}p{1.8cm} >{\raggedright}p{5.75cm} >{\raggedright}p{4cm} >{\raggedright\arraybackslash}p{3.5cm}}
\hline\hline 
{\bf Design} & {\bf Randomization determines} & {\bf Pros} & {\bf Cons} \\\hline 
Mirrored Question & Whether answers sensitive item (``I have the sensitive trait'') or its inverse (``I do not have the sensitive trait'') & Simple implementation & Low respondent confidence in the answer being hidden \\\hline 
Forced Response & Whether answers sensitive item or with forced `yes' or `no' & Simple implementation & Respondents with forced `yes' may fail to say `yes' due to concern that their response might be 
interpreted as an affirmative admission to the sensitive item 
\\\hline 
Disguised Response & Order of red and black cards in two decks of cards. Respondent states the color chosen from the right deck for `yes' to the sensitive item and the color chosen from the left deck for `no' & Best for items where even saying `yes' out loud is sensitive & Complicated randomization device requires in-person implementation \\\hline 
Unrelated Question & Whether answers sensitive item or unrelated,
non-sensitive item & High respondent confidence in the answer being
hidden & The response to the unrelated question must be either independent of respondent characteristics or modeled \\
\hline \hline
\end{tabular}
\caption{Comparison of Four Basic Randomized Response Designs.}
\label{tab:designs}
\end{table}

We now compare the four basic designs of randomized
response technique from the point of view of applied researchers.  
This is summarized in Table~\ref{tab:designs}.  While
both the mirrored question and forced response designs are the
simplest to implement and understand, they have shortcomings.  The
mirrored question design may suffer from low respondent confidence
because both question options -- the sensitive item and its complement
-- are sensitive in nature. Thus, the respondent must 
understand how the method works, whereas a greater degree of random
noise is introduced in the other designs. Confidence may be similarly
reduced in both the mirrored and forced response designs because
although random noise is introduced by the design, the respondent must
still respond `yes' in some circumstances which may still be sensitive
depending on the context.

Statistical power provides another metric for choosing between the 
designs. However, when the design parameters are unconstrained, no
 design dominates any other in terms of statistical power. There 
are some values of $p$, for example, that make the forced 
response design preferable to the mirrored design and other
values of $p$ for which the reverse is true.  Typically, practical
considerations such as the limited availability or suitability of 
certain randomization devices places constraints on the feasible 
values of the design parameters. In such cases, the power
comparisons such as  Figures~\ref{fig:compare:p}~and~\ref{fig:compare:p1} provided in the Appendix may yield preferable designs. 
For example, if the
researcher can only use values of $p$ below $.25$, the mirrored 
question design and the disguised response design dominate 
the forced response design with $p_1 = .1$ or $p_1 = .5$ and 
the unrelated question design with $(1-p)\cdot q = .1$ or 
$(1-p)\cdot q = .5$. There are also randomization devices
that may remove these constraints, such as the use of spinners 
described in \citet{Gingerich:2010}.

Ultimately, the choice of the design can be determined by an 
an assessment of the practical constraints in the research context
and by careful pilot testing of one or more designs. Pilot testing
may help researchers identify the nature of the sensitivity, and
for example point them to the disguised response design because
respondents are unwilling to answer `yes' even with the 
protections of the mirrored question or forced response design.
In addition, the researcher can learn how sensitive the question is
for respondents and use this to determine how much protection 
is needed through the choice of $p$, for example.

\section{Empirical Illustration with the Forced Response Design}
\label{sec:Application}

For empirical illustration, we apply some of the methodologies
proposed above to an original survey we conducted in Nigeria in
2013. A goal of the survey is to estimate the proportion of the
population who knew or came into regular contact with armed
groups. Disclosing social connections with members of armed groups
was tremendously sensitive because it could have put the respondent or
the former armed group member in danger.  When asked such a sensitive
question directly, the respondent would likely refuse to answer
or lie and respond that they had no social connection regardless of
their truthful experience.  All of the analyses in this section are carried 
out with our accompanying open-source software package {\tt rr}.

\subsection{Design}

To address these concerns of non-response and social desirability
bias, we used the forced response design of the randomized response
method.  This technique allowed us to protect the anonymity of each
individual-level response about whether respondents held social
connections to armed groups. Respondents would thus be more willing to
respond honestly to the question. A survey was administered to a
random sample of $2,448$ respondents from 204 communities that
are representative of communities affected by armed militancy from
2007 to 2008. 

The sensitive question, whose question wording appears in
Section~\ref{sec:design:forced}, asked respondents about direct social
connections to armed groups. The forced response design 
was used with a six-sided dice with a dice roll of 2, 3, 4, or 5 corresponding 
to a truthful response ($p=2/3$), a 6 corresponding to a forced `yes'
($p_1=1/6$), and a 1 to a forced `no' ($p_0=1/6$).

\subsection{Power Analysis}

With funds for approximately $2,500$ respondents, we examined the
power of the design to detect a proportion of affirmative responses
ranging from $0\%$ to $15\%$. Using the expression given in
equation~\eqref{eq:powerfunc1}, we examined the power under a range of
other proportions, depicted in Figure~\ref{fig:power}.  For example,
the power of the test to detect a proportion of $10\%$, our prior
expectation, is approximately \Sexpr{power.rr.test(p = 2/3, p1 = 1/6, 
p0 = 1/6, n = 2500, presp = .1, presp.null = 0, design = "forced-known", 
sig.level = .01, type = "one.sample", alternative = "one.sided")$power}.  
Based on this analysis, we concluded that there would be sufficient 
power based on our chosen sample size.

\begin{figure}[t]
\centering
<<powerPlot, fig.width = 5, fig.height = 4>>=
set.seed(1)
presp.seq <- seq(from = 0, to = .15, by = .0025)
n.seq <- c(250, 500, 1000, 2000, 2500)        

power.rr.plot(p = 2/3, p1 = 1/6, p0 = 1/6, n.seq = n.seq, 
              presp.seq = presp.seq, presp.null = 0,
              design = "forced-known", sig.level = .01, 
              type = "one.sample",
              alternative = "one.sided", 
              legend.x = .12, legend.y = .4)

@
\caption{Statistical Power Analysis for the Nigeria Survey Data.  The
  plot displays the power function for detecting varying proportions
  answering the sensitive item in the affirmative (the horizontal
  axis) based on different sample sizes, 250 (lightest line) to 2,500
  (darkest line).}
\label{fig:power}
\end{figure}

\subsection{Multivariate Analysis}

<<label = means>>=
set.seed(1)

p <- 2/3
p1 <- 1/6
p0 <- 1/6

pr.z.q1 <- (mean(nigeria$rr.q1, na.rm = T) - p1 ) / (1 - p0 - p1)

rr.q1.reg.obj <- rrreg(rr.q1 ~ 1, data = nigeria, p = p, p1 = p1, p0 = p0, design = "forced-known")
rr.q1.reg.pred <- predict(rr.q1.reg.obj, avg = T, given.z = F, quasi.bayes = "single", n.sims = 10000)
@

We begin by estimating the proportion of respondents who answer `yes'
to the sensitive question. Based on the observed responses and the
design probabilities, we use equation~\eqref{eq:forced:z} to calculate
the posterior estimate of the proportion of those who had social
connections with a militant. We estimate this proportion to be
\Sexpr{round(rr.q1.reg.pred$est*100)}\% of respondents with a 95\%
confidence interval of \Sexpr{round(rr.q1.reg.pred$ci.lower*100,)} to
\Sexpr{round(rr.q1.reg.pred$ci.upper*100)}\%.

In addition to estimating the proportion of respondents who hold
direct social connections with members of armed groups, it is useful
to examine which types of civilians are connected to the groups. To do
this, we conduct the multivariate regression analysis described in
Section~\ref{subsec:model}. In particular, we predict whether
respondents hold social connections with armed groups as a function
of the assets owned by the respondent (an index of nine assets
including radio, T.V., motorbike, car, mobile phone, refrigerator,
goat, chicken, and cow), marital status ($1=$ married or divorced,
$0=$ single), age, education level (from $1=$ no schooling
to $10=$ post-graduate education), and gender (male or female).  We
use the logistic regression for $f_\beta(x)$ with these covariates as
linear predictors.

<<multivariateSetup>>=
set.seed(1)
rr.q1.reg.obj <- rrreg(rr.q1 ~ cov.asset.index + cov.married + I(cov.age/10) + I((cov.age/10)^2) + cov.education + cov.female, data = nigeria, p = p, p1 = p1, p0 = p0, design = "forced-known")

df.female <- df.male <- nigeria
df.female$cov.female <- 1
df.male$cov.female <- 0
rr.q1.reg.pred.female <- predict(rr.q1.reg.obj, given.y = T, avg = T, newdata = df.female, quasi.bayes = T, n.sims = 10000)
rr.q1.reg.pred.male <- predict(rr.q1.reg.obj, given.y = T, avg = T, newdata = df.male, quasi.bayes = T, n.sims = 10000)
@

\begin{table}[t]
\centering
\begin{tabular}{r..}
  \hline
& \multicolumn{1}{c}{est.} & \multicolumn{1}{c}{s.e.} \\
<<multivariatePrint, results = 'asis'>>=
tab <- data.frame(est = rr.q1.reg.obj$est, se = rr.q1.reg.obj$se)
tab[4,] <- tab[4,] * 10
tab[5,] <- tab[5,] * 100

rownames(tab) <- c("(Intercept)", "Asset Index", "Married", "Age", "Age, Squared", "Education level", "Female")
tab <- tab[c(2:7, 1), ]
colnames(tab) <- c("est", "s.e.")
print(xtable(tab, digits = 3), floating = FALSE, include.colnames = F, only.contents = T)
@
\end{tabular}
\caption{The Estimated Logistic Regression Coefficients from the
  Multivariate Regression Analysis.  The model predicts whether the
  respondent answered the ``self contact'' (with militant groups)
  sensitive item in the affirmative.}
\label{tab:regression}
\end{table}

The estimated coefficients from this model, along with standard
errors, are reported in Table~\ref{tab:regression}.  The results imply
that respondents who have more assets in the household --- including
radios, televisions, refrigerators --- are substantially more likely
to be socially connected to armed groups. Women are
substantially less likely to be connected, while age holds a curvilinear
relationship with militant connections. Marital status and
education levels are not strongly associated with social connections
to armed groups.

We can also compare the predicted probabilities of a `yes' response to
the sensitive item using the fitted model. Based on this logistic
regression model and the individual-level posterior predicted
probability for the forced response design defined in
equation~\eqref{eq:EZpost.forced}, we estimate that
\Sexpr{round(rr.q1.reg.pred.female$est*100)}\% of women shared social
connections with members of armed groups, compared to
\Sexpr{round(rr.q1.reg.pred.male$est*100)}\%
of men with the 95\% confidence intervals of
\Sexpr{round(rr.q1.reg.pred.female$ci.lower*100)} to
\Sexpr{round(rr.q1.reg.pred.female$ci.upper*100)}\%
and
\Sexpr{round(rr.q1.reg.pred.male$ci.lower*100)} to
\Sexpr{round(rr.q1.reg.pred.male$ci.upper*100)}\%, respectively.

\subsection{Using Randomized Response as a Predictor}

Finally, we examine whether people with social connections to armed
groups are more or less likely to join civic groups in their
communities, such as youth groups, women's groups, or community
development committees. We accomplish this by using the methodology 
proposed in Section~\ref{subsec:predict}. We jointly model the probability of
answering ``yes'' to the sensitive item and the probability of joining
a civic group, both using logistic regression with the same set of
predictors.  The outcome model includes the militant contact as the
additional key predictor.

<<predictorSetup>>=

summ <- function(x) return(list(est = mean(x), ci.lower = quantile(x, .025), ci.upper = quantile(x, .975)))

set.seed(44)

rr.q1.pred.obj <- 
    rrreg.predictor(civic ~ cov.asset.index + cov.married + I(cov.age/10) + I((cov.age/10)^2) 
    	      + cov.education + cov.female 
              + rr.q1, rr.item = "rr.q1", parstart = FALSE, estconv = TRUE,
              data = nigeria, verbose = FALSE, optim = TRUE,
              p = p, p1 = p1, p0 = p0, design = "forced-known")
              
rr.q1.predictor.obj.pred.no <- predict(rr.q1.pred.obj, fix.z = 0, avg = T, quasi.bayes = T, keep.draws = T, n.sims = 10000)
rr.q1.predictor.obj.pred.yes <- predict(rr.q1.pred.obj, fix.z = 1, avg = T, quasi.bayes = T, keep.draws = T, n.sims = 10000)
rr.q1.predictor.obj.pred.diff <- summ(rr.q1.predictor.obj.pred.yes$qoi.draws - rr.q1.predictor.obj.pred.no$qoi.draws)

@

\begin{table}[t]
\centering
\begin{tabular}{r....}
  \hline
  & \multicolumn{2}{c}{Joined} &  \multicolumn{2}{c}{Militant } \\
& \multicolumn{2}{c}{Civic Group} &  \multicolumn{2}{c}{ Connection} \\
& \multicolumn{1}{c}{est.} & \multicolumn{1}{c}{s.e.} & \multicolumn{1}{c}{est.} & \multicolumn{1}{c}{s.e.} \\
<<predictorPrint, results = 'asis'>>=
tab <- data.frame(est.outcome = rr.q1.pred.obj$est.t, se.outcome = rr.q1.pred.obj$se.t, est.sens = c(rr.q1.pred.obj$est.b, NA), se.sens = c(rr.q1.pred.obj$se.b, NA))
rownames(tab) <- c("(Intercept)", "Asset Index", "Married", "Age", "Age, Squared", "Education level", "Female", "Militant Connection")
tab[4,] <- tab[4,] * 10
tab[5,] <- tab[5,] * 100

tab <- tab[c(8, 2:7, 1), ]
print(xtable(tab, digits = 3), floating = FALSE, include.colnames = F, only.contents = T)
@
\end{tabular}
\caption{Multivariate Joint Model of Responses to an Outcome
  Regression (``Joined Civic Group'') and a Randomized Response
  Sensitive Item (``Militant Connection''). Estimated coefficients
  from logistic regressions with standard errors are reported in each
  case.}
\label{tab:regression:outcome}
\end{table}

The estimated coefficients from this multivariate joint model are
presented, along with standard errors, in
Table~\ref{tab:regression:outcome}.  The first two columns present the
results from the outcome regression model while the last two columns
show those from the submodel predicting contact with militant groups.
The results suggest that respondents who are socially connected to
armed groups are more likely to later join civic
groups. \Sexpr{round(rr.q1.predictor.obj.pred.yes$est*100, 0)}\% of those who are
connected to armed groups are predicted to join a civic group (a 95\%
confidence interval from \Sexpr{round(rr.q1.predictor.obj.pred.yes$ci.lower*100, 0)} to
\Sexpr{round(rr.q1.predictor.obj.pred.yes$ci.upper*100, 0)}\%), compared to
\Sexpr{round(rr.q1.predictor.obj.pred.no$est*100, 0)}\% of those who are not connected
to the groups (a 95\% confidence interval from
\Sexpr{round(rr.q1.predictor.obj.pred.no$ci.lower*100, 0)} to
\Sexpr{round(rr.q1.predictor.obj.pred.no$ci.upper*100, 0)}\%). The difference is estimated
to be \Sexpr{round(rr.q1.predictor.obj.pred.diff$est*100,0)}\% with a 95\% confidence
interval of \Sexpr{round(rr.q1.predictor.obj.pred.diff$ci.lower*100, 1)} to
\Sexpr{round(rr.q1.predictor.obj.pred.diff$ci.upper*100,0)}\%.

\section{Modified Designs with Unknown Probability}
\label{sec:ModifiedDesigns}

All of the four basic designs explained in
Sections~\ref{sec:Designs}~and~\ref{sec:Statistics} assume that the
randomization distribution is known and respondents comply with the
instruction.  However, such assumptions may be violated in practice.
For example, when surveys are conducted via phone rather than in
person, respondents may not flip a coin as instructed especially if a
coin is not readily available.  As a second consideration, the
unrelated question design requires researchers to know the response
distribution to the unrelated question.  But, this information may not
be available.

In this section, we introduce the two designs, one new and the other 
existing, that allow these probabilities to be estimated 
\citep[see also][for a model-based, rather than design-based, strategy]{Van:2009}. 
In particular, we modify the forced response and unrelated question
designs.  The disadvantage of these modified designs, however, is that 
they require a larger sample size to maintain the same level of statistical 
power.

\subsection{Designs}
\label{subsec:modified}

We consider the forced response and unrelated question designs with
unknown probability.  We show that both of these modified designs are
based on the same identification strategy and hence the identical
estimation method is applicable.

\paragraph{Forced Response Design with Noncompliance.}  
Under the standard forced response design, we assume that both the probability of answering the sensitive question
$p$ and that of a forced `yes' (`no') $p_1$ ($p_0$) are known.  However, respondents may not reply `yes' even when forced to do so \citep{Edgell:1982}.  The modified forced response design addresses such noncompliance behavior.  The assumption here is that sensitive questions lead to under-reporting though a design similar to the one we propose here can also be used to investigate the instances of over-reporting.

Suppose that we set the probability of a forced `no' to zero.  We
first randomly split the sample of respondents into two groups.  In
the first group ($G_i = 1$), respondents are instructed to flip a coin
and answer the sensitive question truthfully if they get heads
($\Pr({\rm heads})=p$).  We assume that this probability $p$ is known
and respondents do provide a truthful answer to the sensitive
question.  If they get tails, then respondents are instructed to
answer `yes' but we allow for some non-compliance.  That is, the
unknown proportion of these respondents may reply `no.'  Let $1-q$
denote the probability of such non-compliance ($q$ is the probability
of compliance).  

In the second group $(G_i = 0)$, respondents truthfully answer the
sensitive question if they obtain tails whereas they are instructed to
reply with `yes' if they get heads.  As in the case of the first
group, we allow for non-compliance and assume that some of the
respondents who are told to say `yes' may answer `no' with the same
probability $1-q$. This assumption holds because the two groups are
randomly sampled.  Then, the resulting estimating equations are given
as follows,
\begin{eqnarray}
  \Pr(Y_i = 1 \mid G_i = 1) &=& p \Pr(Z_i = 1) + (1-p) q \label{eq:modforce1} \\
  \Pr(Y_i = 1 \mid G_i = 0) &=& (1-p) \Pr(Z_i = 1) + p q. \label{eq:modforce2}
\end{eqnarray}
Solving for $\Pr(Z_i = 1)$ yields,
\begin{eqnarray}
  \Pr(Z_i = 1) & = & \frac{1}{2p-1}\l\{p\Pr(Y_i = 1 \mid G_i = 1) -
  (1-p)\Pr(Y_i = 1 \mid G_i = 0) \r\}. \label{eq:modforce}
\end{eqnarray}
Note that it is possible to use different randomization probabilities
for two groups so long as they are known to the researchers.

It is important to note that while this modified design addresses a particular type of non-compliance, in practice respondents may exhibit other types of non-compliance behavior.  This design is a special case of the design proposed by \citet{Clark:1998} who also consider assigning different probabilities for randomization device across groups.  Below, we contribute to this literature by developing a multivariate regression technique and power analysis for this modified design.

\paragraph{Unrelated Question Design with Unknown Probability.}
Under the standard unrelated question design, we assume that the
response probability to the unrelated question is known.  However,
such information may be unreliable or even non-existent.  The
motivation of the modified unrelated question design we consider here
is to assume that this response probability is unknown.  Specifically,
we first randomly split the respondents into two groups.  In the first
group $(G_i = 1)$, the respondents are instructed to flip a coin and
answer the sensitive question if they get heads ($\Pr({\rm heads}) =
p$).  We assume that this probability is known.  The respondents
answer the unrelated question if the outcome of the coin flip is tails. 

In the second group $(G_i = 0)$, we reverse the instructions.
That is, assuming that the coin flip has the same randomization
distribution, if the respondents get heads (tails), they are told to
answer the sensitive (unrelated) question.  This modified design has
been used in the literature.  The applications include studies of drug
use \citep{Goodstadt:1975}, shoplifting \citep{Reinmuth:1975}, voting
\citep{Locander:1976}, and compliance with medication
\citep{Volicer:1982}.

If we let $q$ represent the probability of answering `yes' to the
unrelated question, the estimating equations are identical to those of
the modified forced response design discussed above (i.e.,
equations~\eqref{eq:modforce1}~and~\eqref{eq:modforce2}).  Thus, the
probability of affirmatively answering the sensitive question is also
the same and is given in equation~\eqref{eq:modforce}.
 
\subsection{Multivariate Regression Analysis}
\label{subsec:modregress}

We first consider an approach that has an important advantage of
avoiding the specification of regression function for the unknown
probability $q$.  We base our inference on the following moment
condition derived using the equality given in
equation~\eqref{eq:modforce},
\begin{eqnarray}
  f_\beta(X_i) & = & \frac{1}{2p-1} \{p \Pr(Y_i = 1 \mid
  X_i, G_i = 1) - (1-p) \Pr(Y_i = 1 \mid X_i, G_i = 0) \}. \label{eq:modforceX}
\end{eqnarray}
In this framework, the regression function of interest,
$f_\beta(X_i)$, is obtained as a result of modeling the observed
response, $\Pr(Y_i = 1 \mid X_i, G_i)$.  An obvious disadvantage of
this approach is that one cannot directly specify the latent response
to the sensitive question.  Rather, we obtain the model specification
as a byproduct of the model for the observed response.  For example,
even if we wish to use the logistic regression for $f_\beta(X_i)$, it
is not straightforward to obtain a model for the observed response,
which satisfies equation~\eqref{eq:modforceX}.  One exception is the
linear probability model, $f_\beta(X_i) = \beta^\top X_i$.  In this
case, we can also use linear probability models for $\Pr(Y_i = 1 \mid
X_i, G_i)$ while satisfying equation~\eqref{eq:modforceX}.  Despite
this issue, the proposed approach avoids modeling the unknown
probability $q$ and hence rests on the less stringent assumptions.


The second approach we consider follows the modeling and estimation
strategy outlined in
Sections~\ref{subsec:model}~and~\ref{subsec:estimation} for the
standard designs.  Unlike the previous one, this approach requires
researchers to specify a regression model for the unknown probability
$q = q_\gamma(X_i)$ while allowing them to directly model the latent
response to the sensitive question.  The inference is based on the
following likelihood function,
\begin{equation*}
  \cL(\beta,\gamma \mid \{X_i, Y_i, G_i\}_{i=1}^n) \ = \ \prod_{i=1}^n \{e(G_i)
  f_\beta(X_i) + e(1-G_i) q_\gamma(X_i)\}^{Y_i} [1- \{e(G_i)
  f_\beta(X_i) + e(1-G_i) q_\gamma(X_i)\}]^{1-Y_i} 
\end{equation*}
where $e(G_i) = G_i p + (1-G_i) (1-p)$.  

To maximize this likelihood function, the EM algorithm is useful.
Consider the complete-data likelihood function of the following form, 
\begin{equation}
  \cL_{com}(\beta,\gamma \mid \{X_i, Y_i, G_i, S_i\}_{i=1}^n) \ = \
  \prod_{i=1}^n\l[ f_\beta(X_i)^{Y_i} \{1-
  f_\beta(X_i)\}^{1-Y_i}\r]^{S_i} \l\{q_\gamma(X_i)^{Y_i}
  (1-q_\gamma(X_i))^{1-Y_i}\r\}^{1-S_i} \label{eq:modllik}
\end{equation}
where $S_i$ indicates whether respondent $i$ answers the sensitive
question ($S_i=1$) or not ($S_i = 0$) under each modified design.
Now, we can derive the E-step as follows,
\begin{equation*}
  \E(S_i \mid X_i = x, Y_i = y, G_i = g) 
% & = & \frac{\Pr(S_i = 1 \mid
%    G_i = g, X_i = x)\Pr(Y_i = y \mid S_i = 1, G_i = g, X_i =
%    x)}{\sum_{s=0}^1 \Pr(S_i = s \mid
%    G_i = g, X_i = x)\Pr(Y_i = y \mid S_i = s, G_i = g, X_i =
%    x)} \nonumber \\
  \ = \ \frac{e(g)f_\beta(x)^y
    (1-f_\beta(x))^{1-y}}{e(g)f_\beta(x)^y
    (1-f_\beta(x))^{1-y}+ e(1-g) q_\gamma(x)^y
    (1-q_\gamma(x))^{1-y}}. 
\end{equation*}
Then, the M-step can be implemented by maximizing the following
objective function with respect to $\beta$ and $\gamma$,
\begin{eqnarray*}
  & & \sum_{i=1}^n w_S(X_i, Y_i, G_i)\l\{ Y_i \log f_\beta(X_i) + (1-Y_i)
  \log (1-f_\beta(X_i))\r\}\nonumber \\
  & & \hspace{.5in} + \{1-w_S(X_i, Y_i, G_i)\} \l\{ Y_i \log q_\gamma(X_i) + (1-Y_i)
  \log (1-q_\gamma(X_i))\r\}
\end{eqnarray*}
where $w_S(X_i, Y_i, G_i) = \E(S_i \mid X_i, Y_i, G_i)$.  This
optimization can be easily done by separately fitting two weighted
regressions, $f_\beta(X_i)$ and $q_\gamma(X_i)$.

Although we do not provide details, we can also apply the modeling
strategy similar to the one described in Section~\ref{subsec:predict}
in order to use the latent response to the sensitive question as an
explanatory variable in outcome regression models.


\subsection{Power Analysis}
\label{subsec:modpower}

To conduct power analysis for these modified designs, we first derive
the analytical expression for the standard error.  Akin to section
\ref{subsec:basicpower}, we have, without loss of generality, $f =
f_\beta(1) = \exp(\beta)/ \{1+ \exp(\beta)\}$, which is the
probability of possessing the sensitive trait, and $q = q_\gamma(1) =
\exp(\gamma)/ \{1+ \exp(\gamma)\}$, which is the probability of
possessing the unrelated trait.  Given the likelihood function in
equation~\eqref{eq:modllik}, The Fisher information with respect to
$f$ is given by,
\begin{equation*}
 \mathcal{I}(p,q,r,f) \ = \
   \frac{r p^2}{\{p f + (1-p)q\} \{1- pf - (1-p)q\}} + \frac{(1-r)(1-p)^2}{\{(1-p) f + pq\} \{1- (1-p)f - pq\}}
\end{equation*}
where $r = \Pr(G_i = 1)$ and each term essentially follows the Fisher
information in equation~\ref{eq:basicfisher} with $c = p$ and $d =
(1-p)q$ for $G_i = 1$, and $c = (1 -p)$ and $d = pq$ for $G_i = 0$.
Thus, the standard error of $\hat{f}$ under the modified designs is,
$\sigma(p,q,r,f,n) = 1/\sqrt{n\ \mathcal{I}(p,q,r,f)}$.  Using this
standard error expression, the power functions under one and two-sided
hypothesis tests are identical to those under the basic designs, given
in equations~\eqref{eq:powerfunc1}~and~\eqref{eq:powerfunc2},
respectively.

\subsection{Possible Extensions}

The idea of randomly splitting the sample into two groups can be
applied in variety of ways to make the standard designs robust to a
certain deviation from the assumptions.  We illustrate this by
introducing another modified forced response design.  Under the
standard forced response design, the probability of answering the
sensitive question $p$ as well as the probability of forced `yes' and
`no' responses, $p_0$ and $p_1$ respectively, are assumed to be known.
In Section~\ref{subsec:modified}, we address possible non-compliance
to forced response by allowing some respondents to answer `no' when
they are supposed to say `yes.'  Alternatively, we could assume that
such non-compliance does not exist but the coin flip probability $p$
is unknown.  For example, if the survey is conducted over phone,
respondents may not have access to a coin and hence $p$ may not be
equal to the assumed probability of a coin flip.  Under this
alternative assumption, we have $q=1$ in
equations~\eqref{eq:modforce1}~and~\eqref{eq:modforce2}.  Thus,
solving for $\Pr(Z_i = 1)$ gives, $\Pr(Z_i = 1) = \Pr(Y_i = 1 \mid G_i
= 0) + \Pr(Y_i = 1 \mid G_i = 1) - 1$.  Given this identification
strategy, we can follow the modeling strategies described in
Section~\ref{subsec:modregress} and conduct a multivariate regression
analysis.  We can also derive the power analysis as done in
Section~\ref{subsec:modpower}.

\section{Concluding Remarks}

Since its inception a half century ago, the literature on the
randomized response technique has focused primarily on theoretical
improvements, extensions, and variations in procedures
\citep{Chaudhuri:2011}.  Scholars have assessed the method's
efficiency within various designs \citep[e.g.][]{Moors:1971,
  Dowling:1975, Pollock:1976} and compared them to estimates from
direct questioning \citep[e.g.][]{Lensvelt:2005b, Krumpal:2012,
  Rosenfeld:2014, Gingerich:2014}.  The design originally outlined by
\citet{Warner:1965} has been extended to incorporate multiple
sensitive traits \citep{Abul:1967, Christofides:2005}, multiple
sensitive questions \citep{Raghavarao:1979, Tamhane:1981}, responses
on a Likert scale \citep{Himmelfarb:2008, Dejong:2010}, and
quantitative answers \citep{Eichhorn:1983, Fox:1984}.  Recent work has
explored flexibility in sampling procedure \citep{Chaudhuri:2001,
  Chaudhuri:2005, Chaudhuri:2011}.

While immense methodological progress has been made, the lack of
substantive applications suggests the need for a practical guide
regarding the basic aspects of the randomized response methodology.
In this paper, we describe commonly used designs with examples; show
how to conduct multivariate regression analyses under each design with
the sensitive item as outcome or predictor; develop power analyses;
and propose new designs that address certain deviations from standard
design protocols.  Finally, we offer open-source software to
facilitate the use of these methods.  Taken together, we hope this
paper enables the effective use of the randomized response technique
across disciplines as well as further methodological development.

\newpage
\spacingset{1.4}
\pdfbookmark[1]{References}{References}
\bibliographystyle{natbib}
\bibliography{randresp,my,imai}

\clearpage
\appendix
\pdfbookmark[1]{Appendix: Additional Power Analyses}{Appendix: Additional Power Analyses}
\section*{Appendix: Additional Power Analyses}

\begin{figure}[h]
  \spacingset{1}
<<label = powerComparisonFigureAppendix, fig.height = 7, fig.width = 8, cache = FALSE>>=

linetype <- c("dashed", "dotted", "solid")
              
par(mfrow = c(3,3), oma = c(1.05, 1.5, 2, 0.25), mar = c(2, 3, 0, 0), mgp = c(2, .7, 0), tck = -.015)
              
for(n in c(500, 1000, 2500)) {
      for(z in c(.1, .2, .3)) {
      plot(0,1, type = "n", xlim = c(0, 1), ylim = c(0, 1), axes = F, xlab = "", ylab = "")
              
              p.vals.plot <- c(.2, .4, .6)
              for(j in 1:length(p.vals.plot)){
              
                lines(power.plot(n = n, p = p.vals.plot[j], presp = z, design = "forced-known", vary = "p1"), 
                   lty = linetype[j], lwd = 2)

              }
              
              if(n == 500 & z == .1){
              text(.42, .85, "p = .3", cex = 0.85, pos = 4)
              text(.6, .55, "p = .2", cex = 0.85, pos = 4)
              text(.8, .25, "p = .1", cex = 0.85, pos = 4)
              }
                
              if(n == 500)
              mtext(side = 3, paste("Pr(Z = 1) =", z), cex = .8, line = .75)
              
              if(z == .1)
              mtext(side = 2, paste("N =", n), line = 3.5, cex = .8)
              
              
              axis(1, cex.axis = .8)
              axis(2, las = 1, cex.axis = .8)
              
              mtext(side = 2, "Power", line = 2, cex = 0.7)
              if(n == 2500)
              mtext(side = 1, bquote('Value of p'[1]*' = (1-p)q'), cex = 0.7, line = 2)
              }
        }                   
              @
\caption{Comparison of Power for the Forced Response and Unrelated Question Designs. For three typical values of the probability of a truthful response to the sensitive item ($p$), power is displayed for across values of the probability of a forced `yes' response ($p_1$) for the forced response design and, equivalently, the known proportion of `yes' responses to the unknown question multiplied by the probability of answering the unknown question ($(1-p)q$).}
\label{fig:compare:p1}
\end{figure}


\end{document}


